Nashville is my home airport, and I see people carrying on guitars all of the time. It is very, very common. The FA's will help fit them in the overheads. But, get a hard case even if you have to borrow or rent one. This is an instrument that requires special care.

I usually fly WN, so their policies may be different. I suggest you contact your airline and get a definitive reply on how to proceed. The extra seat is a good idea, but expensive.

You might want to look into shipping the instrument ahead. Take it to FedEx Office (used to be Kinko's) or a UPS store and get a quote. They will have the right size boxes and ask for insurance in the quote. It will probably be cheaper than an extra seat.

For the airplanes you have indicated, there will be room in the overhead, if they will let you use it. Expect people to want to jam their computer bags on top of it. What I have seen on WN (Southwest) is that the guitar goes in first and then the FA will use the remaining space for jackets, hats, purses and other items to open larger spaces in adjacent areas.

Hard side guitar cases, especially fitted ones, are a good investment. A large bass may be hard to fit. You might be able to save a bit for a good travel case by going to a large sporting goods store, Gander Mountain or Bass Pro, and look at large shotgun or compound bow cases. You may be able to adapt one of these for occasional use and it will likely be less expensive than a fitted guitar case.

Whatever you do, do not check it as luggage. Don't even think about it.

Although it contradicts what I have observed, here is the WN policy. Your airline will likely be similar.